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ECOS3010: Tutorial

创建日期：2024-06-03 17:24:27

所属分类：经济Economics

标签： ECOS3010

Tutorial

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ECOS3010: Tutorial 1 (Answer Key)

Question 1-5. Answer True, False or Uncertain. Briey explain your answer.
1. The main di¤erence between at money and commodity money is that at money is
intrinsically useless.
True. Historically, commodity money takes many forms including shell, bead, silver,
gold and etc.. Commodity money has its own value. Unlike commodity money, at money
is intrinsically useless. Fiat money is often issued by the government.
2. The golden rule allocation maximizes the utilities of both the future generations and
the initial old.
False. The golden rule maximizes the utilities of the future generations, but does not
maximize the utilities of the initial old. In our model, the initial old consume only when
old. So they would like allocate all consumption to old. In contrast, all future generations
prefer to consume both when young and when old.
3. In a monetary equilibrium, individuals maximize their utilities subject to the resource
constraints.
False. In a monetary equilibrium, individuals maximize their utilities subject to the
lifetime budget constraints.
4. Our model of money is consistent with the quantity theory of money.
True. Quantity theory of money says that the price level is proportional to the quantity
of money in the economy. In our model, the equality of money supply and money demand
determines the equilibrium price level, which is proportional to the quantity of money.
5. When money supply is constant and population is growing at a constant rate Nt =
nNt1, the allocation from a monetary equilibrium is not the golden rule allocation.
False. When the population is growing and money supply is constant, the allocation
in a monetary equilibrium still achieves the golden rule allocation. One can nd that the
resource constraint coincides with an individuals budget constraint. So individuals in a
monetary equilibrium consume the same consumption bundle as the golden rule allocation
chosen by the planner.
6. Consider an economy with a constant population of N = 100. Individuals are
endowed with y = 20 units of the consumption good when young and nothing when old.
(a) What is the equation for the feasible set of this economy? Portray the feasible set on
a graph. With arbitrarily drawn indi¤erence curves, illustrate the stationary combination
of c1 and c2 that maximizes the utility of future generations.
Feasible set:
100c1 + 100c2 100 20 ! c1 + c2 20:
Point A in on the graph maximizes the utility of future generations.
1
c1
c2
A
c1
*
c2
*
20
20
(b) Now look at a monetary equilibrium. Write down equations that represent the
constraints on rst- and second-period consumption for a typical individual. Combine these
constraints into a lifetime budget constraint.
First-period budget constraint:
c1 + vtmt 20:
Second-period budget constraint:
c2 vt+1mt:
Lifetime budget constraint: using the rst- and second-period budget constraints
c2
vt+1
mt 20 c1
vt
! c1 + vt
vt+1
c2 20:
(c) Suppose the initial old are endowed with a total of M = 400 units of at money.
What condition represents the clearing of the money market in an arbitrary period t? Use
this condition to nd the real rate of return of at money.
Aggregate real demand for money in period t:
N (y c1) = 100 (20 c1):
Aggregate real supply of money in period t:
vtM = 400vt:
The value of money is determined by the equality of money supply and money demand.
Therefore, we have
400vt = 100 (20 c1) and vt = 100 (20 c1)
400
:
Similarly,
vt+1 =
100 (20 c1)
400
:
2
We can now nd that the real rate of return of at money is
vt+1
vt
=
100(20c1)
400
100(20c1)
400
= 1:
The value of money is constant.
Now suppose that preferences are such that u (c1; c2) = c
1=2
1 + c
1=2
2 .
(d) Find an individuals real demand for money. Use the assumption about preferences
and your answer in part (c) to nd an exact numerical value.
In a monetary equilibrium, an individual maximizes his utility subject to the budget
constraint. Mathematically,
max
c1;c2
c
1=2
1 + c
1=2
2 subject to c1 + c2 = 20;
where we have substituted vt=vt+1 in the budget constraint by 1. From the budget con-
straint, c2 = 20 c1. We substitute the expression of c2 into the utility function to have
the unconstrained maximization problem:
max
c1;c2
c
1=2
1 + (20 c1)1=2 :
The rst-order condition is
1
2
c
1=2
1 +
1
2
(20 c1)1=2 (1) = 0
! c1=21 = (20 c1)1=2
! c1 = 20 c1
! c1 = 10:
It follows that an individuals real demand for money is
y c1 = 20 10 = 10:
(e) What is the value of money in period t, vt? What is the price of the consumption
good pt?
The value of money in period t is
vt =
100 (20 c1)
400
=
100 (20 10)
400
= 2:5:
Therefore, the price level is
pt =
1
vt
=
1
2:5
= 0:4:
(f) Suppose instead that the initial old were endowed with a total of 800 units of at
money. How do your answers to part (e) change? Are the initial old better o¤ with more
units of money?
If the initial old is endowed with 800 units of money, it wont a¤ect the choice of (c1; c2).
3
(You can try to verify it.) The value of money in period t is thus
vt =
100 (20 c1)
800
=
100 (20 10)
800
= 1:25:
The price level is
pt =
1
vt
=
1
1:25
= 0:8:
The initial old consume c2 = 10, which is the same as before. So they are not better o¤
with more units of money. In this economy, money is neutral. The change in the stock of
money does not a¤ect any real variables such as c1 and c2. Only nominal variables such as
vt and pt are a¤ected.
7. In this chapter, we modeled growth in an economy by a growing population. We
could also achieve a growing economy by having an endowment that increases over time.
To see this, consider the following economy: Let the number of young people born in each
period be constant at N . There is a constant stock of at money, M . Each young person
born in period t is endowed with yt units of the consumption good when young and nothing
when old. The individual endowment grows over time so that yt = yt1 where > 1.
For simplicity, assume that in each period t, individuals desire to hold real money balances
equal to one-half of their endowment, so that vtmt = yt=2.
(a) Write down equations that represent the constraints on rst- and second-period
consumption for a typical individual. Combine these constraints into a lifetime budget
constraint.
First-period budget constraint:
c1 + vtmt yt :
Second-period budget constraint:
c2 vt+1mt:
As before, we combine the previous two budget constraints to get an individuals lifetime
budget constraint:
c1 +
vt
vt+1
c2 yt:
(b) Write down the condition that represents the clearing of the money market in an
arbitrary period t. Use this condition to nd the real rate of return of at money in a
monetary equilibrium. Explain the path over time of the value of at money.
Aggregate real demand for money in period t:
N (yt c1) :
Aggregate real supply of money in period t:
vtM:
When money market clears, we have
N (y c1) = vtM ! vt = N (yt c1)
M
:
We know that the preferences are such that vtmt = yt=2. It implies that c1 = yt vtmt =
4
yt=2. The value of money in period t is
vt =
N

yt yt2

M
=
Nyt
2M
:
Similarly,
vt+1 =
Nyt+1
2M
:
It follows that the real rate of return of at money is
vt+1
vt
=
Nyt+1
2M
Nyt
2M
=
yt+1
yt
= :
The value of at money grows at a constant rate . In our lecture, we modeled growth in
the economy by growth in the number of young people born each period. We found that in
that case, the rate of return of at money equal to n, the growth rate of the economy. In
this example, is the growth rate of the economy (it is the gross rate of change of the total
endowment.). We discover that even in this more complicated setup, the rate of return of
at money is equal to the growth rate of the economy when the money supply is xed.

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